Constacyclic Codes of Length $2^s$ Over Galois Extension Rings of ${\BBF}_{2}+u{\BBF}_2$

We study all constacyclic codes of length 2<sup>s</sup> over GR<i>(Rfr,m</i>), the Galois extension ring of dimension m of the ring <i>Rfr=</i>F<sub>2</sub>+uF<sub>2</sub>. The units of the ring GR<i>(Rfr,m</i>) are of the forms <i>alpha</i>, and <i>alpha+u</i>beta, where <i>alpha,</i> <i>beta</i> are nonzero elements of F<sub>2</sub>m, which correspond to <i>2</i> <sup>m</sup>(2<sup>m</sup>-1) such constacyclic codes. First, the structure and Hamming distances of <i>(1+u</i>gamma)-constacyclic codes are established. We then classify all cyclic codes of length 2<sup>s</sup> over <i>GR(Rfr,m</i>), and obtain a formula for the number of those cyclic codes, as well as the number of codewords in each code. Finally, one-to-one correspondences between cyclic and <i>alpha</i>-constacyclic codes, as well as <i>(1+u</i>gamma)-constacyclic and <i>(alpha+u</i>beta) -constacyclic codes are provided via ring isomorphisms, that allow us to carry over the results about cyclic and <i>(1+u</i>gamma)-constacyclic accordingly to all constacyclic codes of length 2<sup>s</sup> over <i>GR(Rfr,m</i>).

[1]  S. Berman.  Semisimple cyclic and Abelian codes. II , 1967 .

[2]  Daniel J. Costello,et al.  Polynomial weights and code constructions , 1973, IEEE Trans. Inf. Theory.

[3]  B. R. McDonald.  Finite Rings With Identity , 1974 .

[4]  W. Cary Huffman,et al.  Fundamentals of Error-Correcting Codes , 1975 .

[5]  Ron M. Roth,et al.  On cyclic MDS codes of length q over GF(q) , 1986, IEEE Trans. Inf. Theory.

[6]  A. Nechaev,et al.  Kerdock code in a cyclic form , 1989 .

[7]  Jacobus H. van Lint.  Repeated-root cyclic codes , 1991, IEEE Trans. Inf. Theory.

[8]  James L. Massey,et al.  On Repeated-root Cyclic Codes , 1991, IEEE Trans. Inf. Theory.

[9]  N. J. A. Sloane,et al.  A linear construction for certain Kerdock and Preparata codes , 1993, ArXiv.

[10]  N. J. A. Sloane,et al.  The Z4-linearity of Kerdock, Preparata, Goethals, and related codes , 1994, IEEE Trans. Inf. Theory.

[11]  Cheong Boon Soh,et al.  A note on the q-ary image of a qm-ary repeated-root cyclic code , 1997, IEEE Trans. Inf. Theory.

[12]  Alexis Bonnecaze,et al.  Cyclic Codes and Self-Dual Codes Over , 1999 .

[13]  Parampalli Udaya,et al.  Cyclic Codes and Self-Dual Codes Over F2 + uF2 , 1999, IEEE Trans. Inf. Theory.

[14]  Parampalli Udaya,et al.  Decoding of cyclic codes over F2 + µF2 , 1999, IEEE Trans. Inf. Theory.

[15]  Alexis Bonnecaze,et al.  Decoding of Cyclic Codes Over , 1999 .

[16]  Graham H. Norton,et al.  On the Structure of Linear and Cyclic Codes over a Finite Chain Ring , 2000, Applicable Algebra in Engineering, Communication and Computing.

[17]  Thomas Blackford,et al.  Cyclic Codes Over Z4 of Oddly Even Length , 2001, Discret. Appl. Math..

[18]  Vera Pless,et al.  Type II Codes over , 2002 .

[19]  Irfan Siap.  Linear codes over F2 + u F2 and their complete weight enumerators , 2002 .

[20]  Thomas Blackford.  Cyclic Codes Over Z4 of Oddly Even Length , 2003, Discret. Appl. Math..

[21]  Akihiro Munemasa,et al.  Type II codes over F2 + 11F2 and applications to Hermitian modular forms , 2003 .

[22]  Carmen-Simona Nedeloaia.  Weight distributions of cyclic self-dual codes , 2003, IEEE Trans. Inf. Theory.

[23]  Taher Abualrub,et al.  On the generators of Z4 cyclic codes of length 2e , 2003, IEEE Trans. Inf. Theory.

[24]  Thomas Blackford,et al.  Negacyclic codes over Z4 of even length , 2003, IEEE Trans. Inf. Theory.

[25]  Thomas Blackford,et al.  Negacyclic Codes Over of Even Length , 2003 .

[26]  Taher Abualrub,et al.  On the Generators of Cyclic Codes of Length , 2003 .

[27]  Sergio R. López-Permouth,et al.  Cyclic and negacyclic codes over finite chain rings , 2004, IEEE Transactions on Information Theory.

[28]  Koichi Betsumiya,et al.  Type II codes over F2m+uF2m , 2004, Discret. Math..

[29]  R. Oehmke,et al.  A Mass Formula and Rank of Cyclic Codes of Length , 2004 .

[30]  Ali Ghrayeb,et al.  A mass formula and rank of /spl Zopf//sub 4/ cyclic codes of length 2/sup e/ , 2004, IEEE Transactions on Information Theory.

[31]  Patrick Solé,et al.  On the algebraic structure of quasi-cyclic codes III: generator theory , 2005, IEEE Transactions on Information Theory.

[32]  Mohammed M. AL-Ashker.  Simplex Codes Over the Ring ∑ , 2005 .

[33]  H. Q. Dinh,et al.  Negacyclic codes of length 2/sup s/ over galois rings , 2005, IEEE Transactions on Information Theory.

[34]  AbualrubT.,et al.  On the generators of Z4 cyclic codes of length 2e , 2006 .

[35]  Ana Salagean,et al.  Repeated-root cyclic and negacyclic codes over a finite chain ring , 2006, Discret. Appl. Math..

[36]  Patrick Solé,et al.  On the Algebraic Structure of Quasi-cyclic Codes IV: Repeated Roots , 2006, Des. Codes Cryptogr..

[37]  W. Cary Huffman.  On the decomposition of self-dual codes over F2+uF2 with an automorphism of odd prime order , 2007, Finite Fields Their Appl..

[38]  H. Q. Dinh,et al.  Complete Distances of All Negacyclic Codes of Length $2^{s}$ Over $\BBZ _{2^{a}}$ , 2007, IEEE Transactions on Information Theory.

[39]  Hai Q. Dinh,et al.  Complete Distances of All Negacyclic Codes of Length Over , 2007 .

[40]  Hai Quang Dinh,et al.  On the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions , 2008, Finite Fields Their Appl..

[41]  Taekyun Kim,et al.  A Note on the -Euler Measures , 2009 .

[42]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[43]  P. V. Kumar,et al.  The &-Linearity of Kerdcck , Preparata , Goethals , and Related Codes , 2022 .